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arXiv:1606.05421 · Journal of Physics A Mathematical and General

Why semiclassical electrodynamics is not gauge invariant

Alexander Stewart

Published Dec 7, 2000• 13 citations
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Abstract

It is shown that in semiclassical electrodynamics, which describes how electrically charged particles move according to the laws of quantum mechanics under the influence of a prescribed classical electromagnetic field, only a restricted class of gauge transformations is allowed. This lack of full gauge invariance, in contrast to the situation in classical and quantum electrodynamics which are fully gauge-invariant theories, is due to the requirement that the scalar potential in the Hamiltonian of wave mechanics should represent a physical potential. Probability amplitudes and energy differences are independent of gauge within this restricted class of gauge transformation.

DOI
10.1088/0305-4470/33/50/303

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It is shown that in semiclassical electrodynamics, which describes how electrically charged particles move according to the laws of quantum mechanics under the influence of a prescribed classical electromagnetic field, only a restricted class of gauge transformations is allowed. This lack of full gauge invariance, in contrast to the situation in classical and quantum electrodynamics which are fully gauge-invariant theories, is due to the requirement that the scalar potential in the Hamiltonian of wave mechanics should represent a physical potential. Probability amplitudes and energy differences are independent of gauge within this restricted class of gauge transformation.
The case of a single charged particle with potential energy is considered when the classical electromagnetic field is zero. The vector and scalar potentials describing this situation are not necessarily zero. When the conventional approach to radiation problems using the interaction A⋅p plus p⋅A plus A2 plus A0 (constants omitted) is used, time-dependent probabilities are predicted if the vector potential A=−∇Λ≠0 even though the electromagnetic field is zero. The reason for this unphysical behavior is that eigenfunctions of the unperturbed Hamiltonian, which is not gauge invariant, are used to calculate the probability amplitudes. When a manifestly gauge-invariant procedure is applied to this case the probabilities are constant in all gauges. The reason that physical results are obtained is that eigenstates of the energy operator are used to calculate the probability amplitudes.
The distinction between gauge independence and gauge covariance in electrodynamics is considered for quantities of physical interest. The gauge independence of directly observable quantities, such as the spectra of operators and energy conserving transition rates, is discussed and contrasted with the gauge covariance of other quantities such as scalar products. The quasigauge-invariance of the classical Lagrangian and the invariance of the Euler-Lagrange equations under the addition of a total time derivative to the Lagrangian are discussed. The consequent inherent nonuniqueness of the Lagrangian and Hamiltonian formulations of classical mechanics is pointed out. A physical interpretation of the explicit gauge dependence of classical canonical momenta and of the expectation values of the corresponding quantum mechanical operators is presented. The gauge independence of energy conserving transition rates calculated in the conventional finite-order time dependent perturbation theory is discussed and illustrated. The gauge dependence of the conventional time dependent transition amplitudes in the presence of electromagnetic fields is then discussed, and gauge independent transition amplitudes are constructed. An alternative formulation of the quantum mechanics of charged particles is obtained in terms of a new gauge independent Hamiltonian, which is, as might be expected, unique only within an arbitrary canonical transformation.
A system of nonrelativistic charged particles and radiation is canonically quantized in the Coulomb gauge and Maxwell's equations in quantum electrodynamics are derived. By requiring form invariance of the Schrodinger equation under a space and time dependent unitary transformation, operator gauge transformations on the quantized electromagnetic potentials and state vectors are introduced. These gauge transformed potentials have the same form as gauge transformations in non-Abelian gauge field theories. A gauge-invariant method for solving the time-dependent Schrodinger equation in quantum electrodynamics is given. Maxwell's equations are written in a form which holds in all gauges and which has formal similarity to the equations of motion of non-Abelian gauge fields. A gauge-invariant derivation of conservation of energy in quantum electrodynamics is given. An operator gauge transformation is made to the multipolar gauge in which the potentials are expressed in terms of the electromagnetic fields. The multipolar Hamiltonian is shown to be the minimally coupled Hamiltonian with the electromagnetic potentials in the multipolar gauge. The model of a charged harmonic oscillator in a single-mode electromagnetic field is considered as an example. The gauge-invariant procedure for solving the time-dependent Schrodinger equation is used to obtain the gauge-invariant probabilities that the oscillator is in an energy eigenstate For comparison, the conventional approach is also used to solve the harmonic oscillator problem and is shown to give gauge-dependent amplitudes.
The main notions of semiclassical scalar electrodynamics in different gauges (Hamiltonian, Couloumb, Lorentz) are discussed. These are semiclassical states, Poincare transformations, fields, observables, gauge equivalence. General properties of these objects are formulated as axioms of semiclassical theory; they are heuristically justified. In particular, a semiclassical state may be viewed as a set of classical background field and quantum state in the external background. Superpositions of these "elementary" states can be also considered. Set of all "elementary" semiclassical states forms a semiclassical bundle, with base being classical space and fibres being quantum states in the external background. Quantum symetry transformations (Poincare and gauge transformations) are viewed semiclassically as automorphisms of the semiclassical bundle. Specific features of electrodynamics are investigated for different gauges.
Gauge transformations of the potentials for a prescribed electromagnetic field acting on a non-relativistic charged particle are shown to induce canonical transformations in the extended phase space, in which the time and its conjugate momentum are treated as dynamical variables. The extended Hamiltonian is invariant and the extended canonical equations are covariant under a gauge transformation. The usual time-dependent wave equations in different gauges are obtained from the formalism by using Schrödinger representations.